Structure of Banach algebras over locally compact groups

局部紧群上的 Banach 代数的结构

• 批准号: RGPIN-2014-05354
• 负责人: Samei, Ebrahim
• 金额: $ 0.8万
• 依托单位: University of Saskatchewan
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611586
• 关键词: Structure; Banach; algebras; over; locally

My main field of research is harmonic analysis with an emphasis on non-commutative groups such as amenable groups and Lie groups. Harmonic analysis is a branch of mathematics concerned with the representation of functions, signals, or waves using Fourier series and Fourier transforms. It has become a vast subject with applications in areas such as signal processing and quantum mechanics over the past two centuries. The research projects that I pursue primarily relate to more modern branches of harmonic analysis and is about the understanding of the structure of various Banach algebras related to locally compact groups, including group algebras and Fourier algebras. My objective is to study in a deeper level the structure of these algebras and their relation to the representations of the groups. One aspect of the structure theory of Banach algebras over locally compact groups I am interested in deals with cohomological properties of these algebras. The seminal paper in this respect is certainly B. E. Johnson’s memoir in 1972 in which he introduces and explores the concept of amenability for a Banach algebra. Z-J Ruan’s fundamental work in 1995 of introducing the concept of operator amenability, through the theory of operator spaces, and applying it in non-commutative harmonic analysis has also had a significant influence on this area. An important open problem in this area is the characterization of the weak amenability of the Fourier algebra on non-compact connected groups (the compact and some non-compact cases have already been established). One research project I plan to pursue is to investigate a more general property, namely the failure of spectral synthesis of the antidiagonal for non-abelian, non-compact connected groups. It is expected that this will imply the failure of the weak amenability and provide more such examples. I will also look at the “local behavior” of their Hochschild cohomology. The overall goal is to get more information about the structure of the algebras or the groups using these results. Another research project of mine deals with "non-communicative" analogue of Beurling algebras. These algebras have played important roles in different areas of harmonic analysis. They are L^1-algebras associated to locally compact groups when we put extra "weights" on the groups. With my co-author H. H. Lee, we developed the corresponding "dual theory" and created the Beurling-Fourier algebras (shortly BF algebras). In several co-authored research projects, I have shown that in some aspect, they behave similar to the classical Beurling algebras whereas in others, they behave very differently. However, this theory is very recent and subsequently there is still much research to be completed concerning them. From one direction, more detailed study of BF algebras on compact groups must be conducted. We also need to investigate BF algebras on non-compact, non-abelian groups e.g. Heisenberg groups and the ax+b-group. The research I am planning to pursue using my NSERC Discovery grant is all a continuation of my work in the past 10 years. I am happy to report that the results obtained so far have been substantial and have received positive reviews from other experts in the fields. Canada is growing as one of the research-intensive countries in the world, and to make sure we continue as such, we need to not only maintain our research programs but also grow and expand them. One necessary means, among many others, is to have faculties with strong research initiatives and programs in all areas of science and technologies including mathematics. The research projects I have outlined above will be part of many streams of research being conducted across the country. This will have a great and positive impact in our future.

我的主要研究领域是谐波分析，重点是非共同群体，例如正式的群体和谎言组。谐波分析是使用傅立叶序列和傅立叶变换的功能，信号或波的表示的数学分支。在过去的两个世纪中，它已成为一个广阔的主题，并在信号处理和量子力学等领域中进行了应用。我纯粹与更现代的谐波分析分支有关的研究项目涉及对与本地紧凑型群体有关的各种Banach代数的结构的理解，包括群体代数和傅立叶代数。我的目标是在更深层次的研究中研究这些代数的结构及其与群体表示的关系。 Banach代数的结构理论比本地紧凑的群体对这些代数的共同体特性感兴趣。在这方面的第二篇论文当然是B. E. Johnson的回忆录，1972年他介绍并探讨了Banach代数的舒适性概念。 Z-J Ruan在1995年通过操作员空间理论引入操作员的概念，并将其应用于非交通谐波分析的基本工作也对该领域产生了重大影响。在该领域的一个重要开放问题是傅立叶代数在非紧密连接组上的弱舒适性的表征（已经确定了紧凑型和某些非紧凑型病例）。我计划追求的一个研究项目是调查一个更一般的特性，即抗抗异基对非亚伯利亚，非紧凑型群体的频谱合成失败。可以预期，这将意味着弱舒适性的失败并提供更多这样的例子。我还将研究他们的Hochschild共同体的“本地行为”。总体目标是使用这些结果获取有关代数或组的结构的更多信息。 我的另一个研究项目涉及Beurling代数的“非交流”类似物。这些代数在谐波分析的不同领域发挥了重要作用。当我们在这些组上添加额外的“权重”时，它们是与局部紧凑型组相关的l^1-代数。与我的合着者H. H. Lee一起，我们开发了相应的“双重理论”，并创建了Beurling-tourier代数（不久的是BF代数）。在几个合着的研究项目中，我已经表明，在某些方面，它们的行为与古典代数相似，而在其他方面，它们的行为方式却大不相同。但是，这一理论是最近的，随后仍然有很多关于它们的研究。从一个方向上，必须对紧凑型组的BF代数进行更详细的研究。我们还需要研究非紧凑型，非亚伯群的BF代数，例如海森伯格组和斧头+B组。 我打算使用NSERC Discovery Grant购买的研究是我过去十年来我的工作的延续。我很高兴地报告说，到目前为止获得的结果已经很大，并收到了该领域其他专家的积极评价。加拿大正在成为世界上研究密集型国家之一，并确保我们继续这样做，我们不仅需要维护我们的研究计划，而且还需要发展和扩展它们。除其他许多方面，还必须在包括数学在内的所有科学和技术领域中拥有强大的研究计划和计划。我上面概述的研究项目将是全国许多研究流的一部分。这将对我们的未来产生巨大的积极影响。